# DFT breakdown for effects that depend on phase?

Given the Hohenberg-Kohn theorem, a given ground-state density uniquely defines all ground state properties, because in principle the external potential $V_{ext}$ can be understood as a functional of the ground-state density $n_0(r)$.

Now the Schrödinger equation gives me - again, in principle, the ground-state wave function $\Psi_0(r)$, so if I knew the functional $V_{ext}[n(r)]$, I could go from a ground-state density to the ground-state wavefunction.

This might seem counter-intuitive at first, because the density itself seems to not contain any phase information, but Hohenberg-Kohn says that, in fact, all the information is there (just very well hidden, and not readily extracted).

The question is now: Is the missing phase information contained more in the density or more in the mysterious functional? If it's the latter, I would expect that DFT using approximations to the exchange-correlation functional should break down for phenomena that rely somewhat strongly on (local) phase, such as Anderson localization or the Aharonov-Bohm effect?

• Your intuition is in fact correct. The usual theorems have a loop-hole in terms of thermodynamics. Specifically, although for a finite system one could use any basis (and so a free-electron one is as good as any), in a properly infinite system there could be a phase transition, which makes certain states impossible to reach starting from a de-localised basis. Indeed, strongly correlated phases tend to cause problems with DFT. – genneth May 30 '12 at 3:03